# Changing sign estimate manually

validated readers,

Is someone allowed to manually change sign of an estimate (obtained through a OLS), if this is supported by an underlying theory?

My first idea was that this is basically a manipulation, which forces the estimates to go in the direction we want to; in fact, the sign of the estimate should directly capture the "right" direction of the effect. However, I am starting to think that it might be justified in some cases.

My case is the following. My assumption is that a covariate X has a negative effect on Y, however X is represented by a proxy X1 (which is what eventually is included in the model; X does not appear in the regression). So, this is like saying that X is a function of X1. X1 can either have a negative or a positive effect on X. Thus, the sign of the estimate of B1 is negative (as expected since it proxies the effect of a variable with a negative impact on Y), when the impact of X1 on X is also negative, but it is positive when the impact of X1 on X is positive. In the latter situation, can I put a - in front of B1 and justify in the way I did now (yet more formally)?

I should correct myself. We have not changed the sign of the coefficient, but we changed sign of the observed values for this variable. We have experimental data, and we use proxies for competition on the demand side of the labour market (our variable of interest); we know from the previous literature that the competition has a negative effect on the dependent variable. 2 proxies we used are negatively related to Y, and thus also on competition (the proxy decreases in value, the competition increases, Y decreases). The third proxie we use is a rate and moves instead on the same direction of competition (increase in the rate, increase in the competition), therefore it has a positive effect on Y, which makes it a bad proxy. So, I thought we might use the opposite of this variable to be a proxy for competition, so that we keep the negative relationship between the proxy and the dependent variable. The opposite of this rate (by that I mean: 1-rate) is a measure that actually makes sense.

## 2 Answers

You can also make every coefficient your birthdate if you like, but you can't really call the result something based in statistics, and it's certainly not OLS. You also can't test it.

Making the coefficient the negative of the LS estimate is not remotely justified by the issues you bring up.

1) First let's deal with knowing this parameter is positive:

If there are a priori restrictions on the value of something, you build them into the estimation - but then (for a simple regression, as you appear to be discussing) you'll get a parameter estimate of zero or almost zero, not $-\hat{\beta}$.

2) Now, the "there's a proxy":

If "X1 can either have a negative or a positive effect on X", then it's not really doing the job of a proxy; it sounds like either a mediator or a moderator; at the very least a covariate -- not a proxy, and you would need both in your model. (Further, it seems like that sentence has the effect backward; for the coefficient of X1 to have the "wrong sign" from what it would have if X1 weren't there, wouldn't X need to be affecting X1?)

Secondly, you can't just say "If the situation were thus, it could change the sign" and then change the sign as if the supposition were true. There's actually two separate fallacies here!

a) you have not given any justification for retaining the magnitude of the estimate. Even if something were making the sign change, apart from some very restricted circumstances (circumstances in which you'd do something else!), it won't leave with with the same magnitude.

b) "If condition $A$ were true, outcome $B$ would happen" does not, of itself imply that condition $A$ is true. This looks like the fallacy of argument from ignorance, combined with the fallacy argument from consequences -- that is it looks like you're saying - "we don't know A didn't happen, and we don't like the outcome that we got, so we assume A happened in order to get the desired outcome")

For the kind of argument you'd like to apply to be valid you'd have to show that all the parts of the thing you think may have been going on actually happened. There could be numerous alternate explanations - the theory could be wrong in this situation. The data could be bad. There might be yet other variables that are important but which aren't present. Still other things might have happened. You need to show those didn't happen.

Even if you could show that all of the things you think happened did, and even if you could rule out all the alternative plausible explanations that critics could come up with, people are still going to be saying "why on earth would you use a proxy that you now can be pretty sure will give the wrong sign?"

You'd also still have to show such an effect had the right magnitude that would justify simply multiplying the result by -1 instead of -0.1 or -7.8 or some other number. This would be a difficult task.

• Thank you very much for your reply Glen. What if I explain better? – Fuca26 Feb 27 '13 at 22:06
• Hi Luca - Any additional explanation of what's going on is likely to result in more relevant and useful responses, so certainly, clarify the circumstances. We can't take into account what you don't tell us. – Glen_b Feb 27 '13 at 22:12

Adding to @glen_b 's excellent answer, the only remotely justifiable reason I could see for doing what you are doing is if you knew, somehow, that X1 was perfectly negatively correlated to X. One way this might happen is if X is a difference between two values and X1 is the reverse difference.

Even if this were the case, however, it would be better to just multiply X1 by -1 and get X, and you wouldn't need a proxy in the first place.

So, in short, if you are going to manipulate the coefficients after running the regression, you might as well not run the regression in the first place.

• Both of you are perfectly right. I must apologize thoguh. Actually, we did not manipulate a posteriori the coefficients, what we did was to multiply by -1 the observed values of our proxy (being that proxy a rate, we might even change the rate observations into (100-rate), which is a measure that makes economic sense). Is this wrong? – Fuca26 Feb 28 '13 at 0:13